Near infrared spectrographic instruments are used to provide analysis of materials in order to determine the measurable characteristics of materials such as concentrations of the constituents of the materials or physical characteristics of the materials. For example, near infrared spectrographic instruments are used in agriculture to determine the oil, protein, and moisture content of grain, the fat content of meat, the fat, protein and lactose content of milk and the urea content of milk. In addition, near infrared spectrophotometer are used to analyze blood samples and to analyze pharmaceutical samples. The instruments have also been used to measure physical properties or physical characteristics of materials. For example, the instruments have been successfully used to measure the hardness of wheat.
In typical systems of the prior art, linear regressions are used to estimate analyte concentrations or other measurable characteristics from spectroscopic data, even though the raw data are clearly not linear with respect to concentration. Typically, a mathematical pretreatment is applied to the measured spectral data in an attempt to linearize it. The function log (1/R) is an empirical example of such pretreatment applied to remission measurements. The log (1/R) values can be represented in the equations summing products of the log (1/R) values and weighting coefficients or summing products of derivatives of the log (1/R) values and weighting coefficients. To determine concentrations of the constituents of unknown material, log (1/R) values of a multiplicity of known sample materials similar to the unknown material are measured by the spectrographic instrument. The concentrations of the constituents of the known sample materials are known. From the measurements made on the multiplicity of sample materials, the weighting coefficients of the equations relating the analyte concentrations to the pretreated remission measurements can be determined by multiple regression or partial least squares regression. After the coefficients have been determined, the unknown material can be analyzed by the spectrographic instrument using the coefficients that have been determined from the known sample materials. Transmission measurements are also used to analyze materials in a similar manner.
In modern instruments, the measurements on the samples are made at wavelengths distributed throughout the near infrared spectrum and coefficients and equations relating the measurable characteristics to each measurement are developed by linear regression. While the above technique of analyzing materials have proved to be effective and accurate, there are inaccuracies in the measurements which occur in part because the mathematical pretreatment applied to the data doesn't perfectly linearize the data. The assumption of linear regression is that absorption fractions determined for the material relate directly to the concentrations of the absorbers in the material. In real material samples, physical factors, such as inhomogeniety, packing density, and particle size, affect the relationship between the absorption fractions determined for the sample and the concentrations of absorbers in the sample. To the extent these factors vary from sample to sample, increased uncertainty in concentration estimates results compared to the case where their variation is absent. The error caused by these factors is often small compared to other sources of error but cases in which packing or particle size variations visibly affect spectra are frequently encountered. Accordingly, there has been an effort to generate functions which describe the system of reflection from and transmission through the material theoretically and then use these functional forms as a guide in setting up regression analysis. One such theoretical description is the widely used Kubelka-Munk equation, which has a theoretical basis and provides a linear function for the remission function. The Kubelka-Munk equation for remission is: EQU F(R)=(1-R).sup.2 /2R=K/S=2k/2b
In this equation, R is the remission fraction from a sample of infinite thickness, and F(R) has been called the remission function. K and S are the Kubelka-Munk absorption and scattering coefficients. The linear coefficients for absorption and remission are given by k and b.
The Kubelka-Munk theory was developed for dense materials. They drew from work on a rather different kind of system which was examined in 1905 by Schuster in an article entitled "Radiation Through a Foggy Atmosphere". A. Shuster, Astrophysical J. 21, 1 of (1905). In summarizing the application of this work to reflectance spectroscopy, Kortum describes remission as a function of absorption and scattering coefficients. G. Kortum, Reflectance Spectroscopy (Springer-Verlag, New York, N.Y. 1969.) Using the reported definitions for the coefficients, the following equation is obtained: EQU F(R)=2.alpha./(.alpha.+.sigma.)!/.sigma./ (.alpha.+.sigma.)!=2.alpha./.sigma.
This equation is referred to as the Shuster-Kortum equation. Kortum terms .alpha. and .sigma. as the "true absorption" and scattering coefficients of "single scattering". Alternatively .alpha. and .sigma. can be interpreted to be the relative probabilities (fractions) of absorption and scattering respectively for the light which interacts with a single particle.
The above equations, and modifications thereof presented by several other authors, were derived using continuous functions and are collectively referred to as continuum theories. It is believed that the continuum theories are not completely satisfactory, since the scattering coefficient and absorption coefficient refer to units of thickness of homogeneous layers treated as a continuum whereas most material samples are not homogeneous and typically are made up of particles.
Discontinuum theory, sometimes called the statistical method, is quite distinct from the continuum theories in the sense that an actual summation is carried out over the fundamental units of which the absorbing and light-scattering sample is composed. Heretofore, discontinuum theory had limited application because it was applied only to a specific model for the particular sample under consideration. However, it has been recognized that such methods can provide a relationship between the fundamental constants of the particles and the measurable quantities. In this invention, this benefit is extended to a more general case.